Can Blender render pngs with the background transparent?
Yes this is possible.

To save rendered images with transparent background (PNG format for instance), the settings are as follows for "Cycles" and  "Blender Render" rendering engines.

Cycles

Properties Editor > Render Context > Output Panel

Choose RGBA so that the alpha from the renderer is saved in the png

Properties Editor > Render Context > Film Panel

Check Film>Transparent so that the World background is transparent.





Blender Render

Properties Editor > Render Context > Output Panel

Choose RGBA so that the alpha from the renderer is saved in the png

Properties Editor > Render Context > Shading Panel

Choose Transparent Alpha type.



EEVEE (Blender 2.8)

Properties Editor > Render Context > Film Panel

Choose Transparent Alpha type.